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Ultraspherical

Jacobi polynomial with α=β>−1\alpha=\beta>-1, are called the ultraspherical polynomials or Gegenbauer polynomial. We will denote ultraspherical polynomial by Pnλ(x)P_{n}^{\lambda}(x) where, α=λ−12\alpha=\lambda-\frac{1}{2}. Note that λ>−12\lambda>-\frac{1}{2}.

Pnλ(x)=Γ(λ+12)Γ(2λ)Γ(n+2λ)Γ(n+λ+12)Pnα,α(x),α=λ−12P_{n}^{\lambda}(x)=\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma\left(2\lambda\right)}\frac{\Gamma\left(n+2\lambda\right)}{\Gamma\left(n+\lambda+\frac{1}{2}\right)}P_{n}^{\alpha,\alpha}(x),\quad\alpha=\lambda-\frac{1}{2}

we can also write

Pnλ(x)=Γ(α+1)Γ(2α+1)Γ(n+2α+1)Γ(n+α+1)Pnα,α(x),α=λ−12P_{n}^{\lambda}(x)=\frac{\Gamma\left(\alpha+1\right)}{\Gamma\left(2\alpha+1\right)}\frac{\Gamma\left(n+2\alpha+1\right)}{\Gamma\left(n+\alpha+1\right)}P_{n}^{\alpha,\alpha}(x),\quad\alpha=\lambda-\frac{1}{2}

  • Note that for α=−1/2\alpha=-1/2, or λ=0\lambda=0, Γ(2α+1)\Gamma\left(2\alpha+1\right)is not defined, and this case should be handled carefully.

  • The PnλP_{n}^{\lambda} are orthogonal with respect to the following weight function.

w(x)=(1−x2)α=(1−x2)λ−0.5w(x)=(1-x^{2})^{\alpha}=(1-x^{2})^{\lambda-0.5}
  • The support of ultraspherical polynomial is [−1,1].[-1,1].

Methods​